Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-x-y &= 1 \\ 5x-7y &= 4\end{align*}$
Answer: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $5$ and the bottom equation by $1$ $\begin{align*}-5x-5y &= 5\\ 5x-7y &= 4\end{align*}$ Add the top and bottom equations. $-12y = 9$ Divide both sides by $-12$ and reduce as necessary. $y = -\dfrac{3}{4}$ Substitute $-\dfrac{3}{4}$ for $y$ in the top equation. $-x+ \dfrac{3}{4} = 1$ $-x+\dfrac{3}{4} = 1$ $-x = \dfrac{1}{4}$ $x = -\dfrac{1}{4}$ The solution is $\enspace x = -\dfrac{1}{4}, \enspace y = -\dfrac{3}{4}$.